Completing the square

1. Completing the square An alternative approach.

2. To complete the square we are required to write ax2 + bx +c in the form • a(x + p)2 + q. • We will do this by expanding the second expression and comparing coefficients.

3. a = 1 • Example 1 • Write x2 + 6x – 2 in the form • (x + p)2 + q • Start by expanding the bracket. • (x + p)2 + q =

4. Example 1 (continued) • x2 + 2px + p2 + q • Compare x2 + 6x - 2 • From the coefficient of x we can see that • 2p = 6 and so p = 3.

5. Example 1 (continued) • Now we can substitute for p and calculate q. • From the constant term • p2 + q = -2 • so 32 + q = -2 • 9 + q = -2 • q = -11 • giving x2 + 6x – 2 = (x + 3)2 - 11

6. Example 2 • Write x2 – 3x + 1 in the form • (x + p)2 + q. • Again start by expanding the bracket: • (x + p)2 + q =

7. Example 2 (continued) x2+2px + p2 + q Compare x2–3x + 1 From the coefficient of x we see that 2p = -3 and so p = -3/2 Substituting for p to calculate q gives (-3/2)2 + q = 1

8. Example 2 (continued) So that x2 – 3x + 1 =

9. a > 1 or a < 1 • The important difference this time is in the expansion of • a(x + p)2 + q. • We get an extra factor of a in the first three terms; • ax2 + 2apx + ap2 + q

10. Example 3 • Express 3x2 + 12x – 2 in the form a(x + p)2 + q. • Start by expanding the bracket: • a(x2 + 2px + p2) + q which becomes

11. Example 3 (continued) • ax2 + 2apx + ap2 + q • Compare 3x2 + 12x - 2 • We see that a = 3 • 2ap = 12 • 6p = 12 • giving p = 2

12. Example 3 (continued) • Finally from the constant term • ap2 + q = -2 • 3 x 22 + q = -2 • 12 + q = -2 • q = -14 • 3x2 + 12x – 2 = 3(x+2)2 - 14

13. Example 4 • Express 4x2 + 12x – 1 in the form a(x + p)2 + q. • Expand the bracket: • a(x + p)2 = ax2 + 2apx + ap2 + q • Compare 4x2 + 12x - 1

14. Example 4 (continued) • We see that a = 4 • From the middle term we see that 2ap = 12 • 8p = 12 • p = 1.5 or 3/2 • Finally ap2 + q = -1

15. Example 4 (continued) • 9 + q = -1 • q = -10 • So 4x2 +12x – 1 = 4(x+3/2)2 - 10

16. Example 5 • Complete the square for • 3 + 8x – 2x2 • We can swap the order around this time, and make it • q + a(x + p)2

17. Example 5 continued • q + a(x+p)2 = • q + ax2 + 2apx + ap2 • q + ap2 + 2apx + ax2 • Compare 3 + 8x - 2x2 • This shows that a = -2

18. Example 5 continued • 2ap = 8 so • -4p = 8 and p = -2 • q + ap2 = q + (-2) x (-2)2 • 3 = q – 8 • q = 11 • 3 + 8x – 2x2 = 11 – 2 (x - 2)2